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Those Fascinating Numbers
Page15(35 of 451)

Those Fascinating Numbers 15 39 • the smallest number n such that 2n − 7 is prime (a question raised by Erd˝os in 1956): using a computer, one obtains that the other numbers n 20 000 satisfying this property are 715, 1983, 2319, 2499, 3775 and 12819.19 40 • the smallest solution of equation σ(n) n = 9 4 ; the sequence of numbers satisfying this equation begins as follows: 40, 224, 174 592, 492 101 632, . . . 41 • the largest odd number which is not the sum of four non zero squares (Sierpinski [185], p. 404); • the largest number n such that the polynomial x2 + x + n is prime for each of the numbers x = 0, 1, 2, . . . , n − 2; the other numbers n satisfying this property are n = 1, 2, 3, 5, 11 and 17 (see D. Fendel & R.A. Mollin [80]); • the integer part of the number γ0 = 41.677647, that is the conjectured value of lim supn→∞ σ∞(n) log n , where σ∞(n) stands for the smallest number k such that f k(n) = 1, where f(n) = ⎧ ⎨ ⎩ 1 if n = 1, n/2 if n is even, 3n + 1 if n is odd, f 1(n) = f(n), f 2(n) = f(f(n)), f 3(n) = f(f 2(n)) and so on; according to the Syracuse conjecture (also called the 3x + 1 problem ), this sequence inevitably reaches the number 1 (see J.C. Lagarias & A. Weiss [120]); • the smallest prime number of the form (x4 +y4)/2: here 41 = (34 +14)/2: there exist only eight prime numbers 10 000 satisfying20 this property, namely 41, 313, 353, 1201, 3593, 4481, 7321 and 8521. 42 • the smallest number n 1 such that σ2(n) is a perfect square: the sequence of numbers satisfying this property begins as follows: 42, 246, 287, 728, 1434, 1673, 1880, 4264, 6237, 9799, 9855,. . . ; • the fifth Catalan number (see the number 14); 19While performing this search, one can ignore all even numbers n, all numbers n ≡ 1 (mod 4), all numbers n ≡ 7 (mod 10) as well as all numbers n ≡ 11 (mod 12), since in these four cases, we obtain respectively that 3, 5, 11 and 13 divide 2n − 7. 20It is interesting to mention that there exist other forms which generate infinitely many prime numbers: it is the case, for instance, for the form x2 + y4 as it was proved by J. Friedlander & H. Iwaniec [85], as well as for the form x3 + 2y3 as was established by D.R. Heath-Brown [110] (see also for that matter the number 3 391).